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The confidence interval for a mean from a normal distribution when the population standard deviation is known is:

The margin of error is

To solve for n:

The confidence interval for a mean from a normal distribution when the population standard deviation is unknown is:

The margin of error is

To solve for n, calculate the minimum n such that:

Term | Description |
---|---|

sample mean | |

z _{α/2} | inverse cumulative probability of the standard normal distribution at 1- α /2; α = 1 - confidence level/100 |

σ | population standard deviation (assumed known) |

n | sample size |

ME | margin of error |

t _{ α/2} | inverse cumulative probability of a t distribution with n-1 degrees of freedom at 1-α/2 |

S | planning value |

The interval (*P _{L}*,

To solve for n, calculate the minimum n such that:

(P – P_{L}) ≤ ME and (P_{U} – P) ≤ ME where P = planning value proportion.

Term | Description |
---|---|

v_{1} (lower limit) | 2x |

v_{2} (lower limit) | 2(n – x + 1) |

v_{1} (upper limit) | 2(x + 1) |

v_{2} (upper limit) | 2(n – x) |

x | number of events |

n | number of trials |

F (lower limit) | lower α/2 point of F distribution with v_{1} and v_{2} degrees of freedom |

F (upper limit) | upper α/2 point of F distribution with v_{1} and v_{2} degrees of freedom |

The lower bound confidence limit for a rate or mean from a Poisson distribution is:

The upper bound confidence limit for a rate or mean from a Poisson distribution is:

The lower margin of error equals −1 × (lower bound confidence limit). The upper margin of error equals the upper bound confidence limit.

To solve for *n*, calculate the minimum *n* such that:

(*S* – *S*_{L}) ≤ ME and (*S*_{U} – *S*) ≤ ME

Term | Description |
---|---|

n | sample size |

t | length of observation; for the Poisson mean, length = 1 |

s | total number of occurrences in a Poisson process |

χ^{2}_{p, x} | upper x percentile point of a chi-square distribution with p degrees of freedom, where 0 < x < 1 |

S | planning value |

ME | margin of error |

The lower bound confidence limit for variance from a normal distribution is:

The upper bound confidence limit for variance from a normal distribution is:

To obtain the confidence interval for the standard deviation, take the square root of the above equations.

The lower margin of error equals −1 × (lower bound confidence limit). The upper margin of error equals the upper bound confidence limit.

To solve for n for variance, calculate the minimum n such that:

(*S*^{2} – *S*^{2}_{L}) ≤ ME and (*S*^{2}_{U} – *S*^{2}) ≤ ME

To solve for n for standard deviation, calculate the minimum n such that:

(*S* – *S*_{L}) ≤ ME and (*S*_{U} – *S*) ≤ ME

Term | Description |
---|---|

n | sample size |

s^{2} | sample variance |

Χ^{2} _{p} | upper 100p^{th} percentile point on a chi-square distribution with (n – 1) degrees of freedom |

S | planning value |

ME | margin of error |